March 20, 2019

Correlation to Causation

Plan for Today:

(1) Recap

  • correlation
  • two problems:
    • confounding/bias
    • random association

(2) Solutions for Bias

  • design-based vs. adjustment-based
  • conditioning
    • what is it?
    • how does it work?

Correlation to Causation

Testing Causal Claims

  1. We start with a causal claim

  2. Turn claim into a causal theory
    • causal logic, independent \(X\)/dependent \(Y\) variables
  3. Turn causal logic and \(X\)/\(Y\) into hypotheses
    • expect that potential outcomes of \(Y\) change with \(X\)
  4. B/C of FPCI: imperfectly test hypotheses using correlation of observed values of \(X\) and \(Y\)

  5. Infer causality if:
    • assumptions about cases we compare let us ignore confounding/bias
    • correlation unlikely to have occured by chance

Problems with Correlation

Bias/Confounding

Equivalently:

  1. Observed cases with different values of \(X\) have different potential outcomes of \(Y\)
  2. Observed cases with different values of \(X\) have different values of \(W\), which is related to \(Y\)
  3. There is a "backdoor" causal path from \(X\) to \(Y\)

Random Error

  1. Correlation between \(X\) and \(Y\) is by chance and reflects no change in \(Y\) due to \(X\).

Solutions to Problems

Bias/Confounding

  1. adjustment-based solutions
  2. design-based solutions

Random Error

  1. Increase number of independent cases
  2. Take courses in statistics

Solutions to Bias

adjustment-based

  • Identify possible confounding variables (e.g. \(W, Z, V, U\))
  • Measure these variables
  • adjust correlation of \(X\) and \(Y\) by "conditioning" on confounding variables

design-based

  • Compare cases that, by assumption, are
    • similar in terms of confounding variables \(W\)/ potential outcomes of \(Y\)
    • exposed to \(X\) in a manner unrelated to \(W\)/potential outcomes of \(Y\)

Adjustment-Based Solutions

Intuition: Adjustment

Adjustment-based approaches start from the…

Comparative Method:

Sometimes called the "method of difference" (via John Stuart Mill), this assesses whether \(X\) causes \(Y\) by…

  • comparing two cases that are the same in all relevant respects, except for value of \(X\)
  • assess for these two cases whether \(Y\) changes when \(X\) changes (correlation)

Intuition: Adjustment

Question:

What causes the spread of cholera?

Causal claim:

Contaminated water causes cholera outbreaks

Obvious?

19th Century London saw repeated outbreaks of cholera, with mass death

  • Dominant view was that "miasmas" or bad air caused diseases like cholera

Intuition: Adjustment

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John Snow, MD

Intuition: Adjustment

Intuition: Adjustment

Snow could compare places with contaminated water with those that did not, but there is a possibility of confounding:

  • rotting organic material \(\to\) miasmas \(\to\) cholera?
  • rotting organic material \(\to\) contaminated water \(\to\) cholera?

Intuition: Adjustment

Intuition: Adjustment

Comparing observed water quality and cholera in:

  • two different areas with different water quality
  • same area at different times with different water quality

would not rule out miasma as explanation for cholera.

  • "miasma" changes as "water" changes; might affect cholera; \(\to\) confounding

Broad Street Pump Outbreak (1854)

Broad Street Pump Outbreak (1854)

Near the outbreak

Brewers Broad St. Residents
Water Source Brewery Well/
Beer (Clean)
Pump (Contam.)
Location Near pump Near pump
Timing Aug. 1854 Aug. 1854
Miasmas? Yes? Yes?
Cholera No Yes

Broad Street Pump Outbreak (1854)

Far from outbreak

Lady and Niece Non-Soho Residents
Water Source Broad Street Pump
(Contam.)
Another Pump
(Clean)
Location Far from Broad St. Far from Broad St.
Timing Aug. 1854 Aug. 1854
Miasmas? No No
Cholera Yes No

Intuition: Adjustment

Conditioning

A more general approach to the comparative method is:

conditioning

when we observe \(X\) and \(Y\) for multiple cases, we examine the correlation of \(X\) and \(Y\) within groups of cases that have the same values of confounding variables \(W, Z, \ldots\).

How does conditioning solve the problem? Equivalently:

  • Cases compared have same potential outcomes of \(Y\) for different values of \(X\)
  • Cases compared have SAME values on confounding variables \(W\); \(W\) cannot be related to \(X\) or to \(Y\) \(\to\) no confounding
  • "Backdoor" path from \(X\) to \(Y\) is "blocked"

Conditioning

Sometimes we think about "conditioning" like this:

  • conditioning lets us find the correlation between \(X\) and \(Y\), ceteris parabis, "all else being equal".

But all else does not need to be equal:

  • We only need to compare cases that are the same on confounding variables
  • same on variables that are causally linked to \(X\) and \(Y\)

Conditioning: Example

Sanctuary Cities

  • Trump Administration intensified immigration enforcement
  • Some Cities/Counties opted to not assist US Immigrations and Customs Enforcement (ICE) by detaining immigrants for longer
  • These are "Sanctuary Cities"

Effect on Crime

  • Trump and political allies suggest "sanctuary" policies undermine law enforcement, lead to more crime

Does having "sanctuary" policy increase crime?

Conditioning: Example

\(i\) \(Sanctuary_i\) \(Crime_i^{Sanct.}\) \(Crime_i^{No \ Sanct.}\) \(S_i - No \ S_i\)
1 \(\mathbf{Yes}\) \(\mathbf{35}\) 40 -5
2 \(\mathbf{Yes}\) \(\mathbf{35}\) 40 -5
3 \(\mathbf{Yes}\) \(\mathbf{5}\) 10 -5
4 \(\mathbf{No}\) 35 \(\mathbf{40}\) -5
5 \(\mathbf{No}\) 5 \(\mathbf{10}\) -5
6 \(\mathbf{No}\) 5 \(\mathbf{10}\) -5

Conditioning: Example

If we simple look at the unadjusted difference in crime in sanctuary vs. non-sanctuary counties, we would get:

\[\frac{\overbrace{35 + 35 + 5}^{\text{Mean Crime in Sanctuary}}}{3} - \frac{\overbrace{40 + 10 + 10}^{\text{Mean Crime w/out Sanctuary}}}{3} = \frac{15}{3} \neq -5\]

This is bias and must result from confounding

Conditioning: Example

\(Urban_i\) is a confounding variable. Why?

\(i\) \(Sanctuary_i\) \(Urban_i\) \(Crime_i^{Sanct.}\) \(Crime_i^{No \ Sanct.}\) \(S_i - No \ S_i\)
1 \(\mathbf{Yes}\) \(\mathbf{Yes}\) \(\mathbf{35}\) 40 -5
2 \(\mathbf{Yes}\) \(\mathbf{Yes}\) \(\mathbf{35}\) 40 -5
3 \(\mathbf{Yes}\) \(\mathbf{No}\) \(\mathbf{5}\) 10 -5
4 \(\mathbf{No}\) \(\mathbf{Yes}\) 35 \(\mathbf{40}\) -5
5 \(\mathbf{No}\) \(\mathbf{No}\) 5 \(\mathbf{10}\) -5
6 \(\mathbf{No}\) \(\mathbf{No}\) 5 \(\mathbf{10}\) -5

Conditioning: Example

Conditioning: Example

If we condition on \(Urban\) (compare \(X\) and \(Y\) for cases with same value of \(Urban\)), we can remove the bias.

\(i\) \(Sanctuary_i\) \(Urban_i\) \(Crime_i^{Sanct.}\) \(Crime_i^{No \ Sanct.}\) \(S_i - No \ S_i\)
1 \(\mathbf{Yes}\) \(\mathbf{Yes}\) \(\mathbf{35}\) 40 -5
2 \(\mathbf{Yes}\) \(\mathbf{Yes}\) \(\mathbf{35}\) 40 -5
4 \(\mathbf{No}\) \(\mathbf{Yes}\) 35 \(\mathbf{40}\) -5
3 \(\mathbf{Yes}\) \(\mathbf{No}\) \(\mathbf{5}\) 10 -5
5 \(\mathbf{No}\) \(\mathbf{No}\) 5 \(\mathbf{10}\) -5
6 \(\mathbf{No}\) \(\mathbf{No}\) 5 \(\mathbf{10}\) -5

Conditioning: Example

If we condition on \(Urban\), we get the true effect of \(-5\):

\[\frac{\overbrace{35 + 35}^{\text{Sanctuary (Urban)}}}{2} - \frac{\overbrace{40}^{\text{w/out Sanctuary (Urban)}}}{1} = -5\]

\[\frac{\overbrace{5}^{\text{Sanctuary (Not Urban)}}}{1} - \frac{\overbrace{10 + 10}^{\text{w/out Sanctuary (Not Urban)}}}{2} = -5\]

Conditioning: Example

Why does this work?

When we condition on \(W\) (Urban), by looking at cases where \(W\) is the same:

  • there can be no relationship between \(W\) and potential outcomes of \(Y\) (Crime)
  • there can be no relationship between \(W\) and \(X\) (sanctuary status)

Without these two relationships, confounding does not happen, there is no bias.

Conditioning: Example

Conditioning: Next Time

What assumptions must we make

in order for adjustment solutions to confounding/bias to work?